101成大資工程設[參考解答]

一、Data Structure (50%)

1. (20%) Let $S_n$ be the expected number of comparison in a successful search of a randomly constructed n-node binary tree, and let $U_n$ be the expected number of comparisons in an unsuccessful search. We assume that $H_n$ is the n-th harmonic number. Please represent $S_n$ and $U_n$ respectively using harmonic number.

參考解答：

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2. (30%) For the AOE (Activity on Edge) network described by the table, (a) What is the earliest time the project can finish? (b) Please list all critical paths. Note that state 1 is the starting state and state 10 is the goal state.

Activity	From state	To state	Time
$a_1$	1	2	5
$a_2$	1	3	5
$a_3$	2	4	3
$a_4$	3	4	6
$a_5$	3	5	3
$a_6$	4	6	4
$a_7$	4	7	4
$a_8$	4	5	3
$a_9$	5	7	1
$a_{10}$	5	8	4
$a_{11}$	6	10	4
$a_{12}$	7	9	5
$a_{13}$	8	9	2
$a_{14}$	9	10	2

參考解答：
(a) 22
(b)
1: $a_2 \rightarrow a_4 \rightarrow a_8 \rightarrow a_9 \rightarrow a_{12} \rightarrow a_{14}$
2: $a_2 \rightarrow a_4 \rightarrow a_8 \rightarrow a_{10} \rightarrow a_{13} \rightarrow a_{14}$
3: $a_2 \rightarrow a_4 \rightarrow a_7 \rightarrow a_{12} \rightarrow a_{14}$

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二、Algorithms (50%)

1. (20%) Solving the recurrence $T(n) = 2T(n/4) + \sqrt{n}$ using $\Theta$ notation.

參考解答：$\Theta (\sqrt{n} lgn)$

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2. (20%) The incident matrix of a directed graph $G=(V,E)$ with no self-loops is a $|V| \times |E|$ matrix $B = (b_{ij})$ such that

$$
\begin{equation}
b_{ij} =
\begin{cases}
-1 & \text{if edge $j$ leaves vertex $i$}\\
1 & \text{if edge $j$ enters vertex $i$}\\
0 & \text{otherwise}
\end{cases}
\end{equation}
$$

Describe what the entries of the matrix product $BB^T$ represent, where $B^T$ is the transpose of $B.$

參考解答：
Let $A = BB^T = (a_{ij})$
$$
\begin{equation}
a_{ij} =
\begin{cases}
v_i \text{‘s } indeg() + outdeg() & i = j\\
-1 \times \text{edges between } v_i \text{ and } v_j & i \neq j
\end{cases}
\end{equation}
$$

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3. (10%) The Fibonacci numbers are defined by recurrence

$F_0 = 0,$
$F_1 = 1,$
$F_i = F_{i-1} + F_{i-2}$ for $i \geq 2.$
Give an $O(n)$-time dynamic-programming algorithm to compute the nth Fibonacci number.

參考解答：

array f[0...n];

f[0] = 0;

f[1] = 1;

for i = 2 to n:

    f[i] = f[i - 1] + f[i - 2];

return f[n];

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題目(pdf)：連結

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